President Recep Tayyip Erdogan on Sunday called on the EU to be "impartial" in the growing row over energy prospecting in contested eastern Mediterranean waters, his office said.
In a phone call with Charles Michel, the European Council president in turn stressed the importance of de-escalating the conflict and called on Turkey to refrain from activities that fuel the tensions with rival Greece, an EU official said after the talks.
Turkey's hunt for gas and oil reserves in waters claimed by EU member Greece has severely strained the relationship between the two Nato members and pitted Ankara against the EU bloc.
The escalating conflict is set to top the agenda at a September 24-25 European Council meeting, with some member states pressing for sanctions against Turkey.
During the call with Mr Michel, the Turkish leader "invited EU institutions and member states to be fair, impartial and objective and to act responsibly on regional issues, particularly the eastern Mediterranean," the presidency said.
Turkey last month deployed its exploration vessel Orus Reis in waters between Greece and Cyprus, which prompted Athens to stage naval exercises to defend its maritime territory.
Mr Erdogan on Saturday raised the stakes and warned Greece: "They will either understand the language of politics and diplomacy, or on the field through bitter experiences."
The European Union's diplomatic chief Josep Borrell has mooted sanctions against Ankara at the coming meeting but so far France, which sent frigates and fighter jets to the region in support of Greece, has been unable to persuade other EU nations to join its hardline response.
Mr Michel told Mr Erdogan that "all measures [sticks and carrots] will be considered" at this month's meeting, according to the EU official.
And Mr Erdogan said the EU's approach to the issue would be "a test of sincerity" for international law and regional peace, adding that provocative steps taken by some European politicians would not foster a solution, according to the Turkish presidency.
As the tension ran high, Turkey's armed forces on Sunday began annual exercises called "Mediterranean Storm" in the breakaway republic of northern Cyprus – an entity recognised only by Ankara.
"The security priorities of our country and the TRNC (Turkish Republic of Northern Cyprus) are indispensable," Turkish Vice President Fuat Oktay said on Twitter.
The Turkish defence ministry also said the exercises, which are due to last until Thursday, continued "successfully".
Cyprus is divided between the Greek Cypriot-run south – an EU member state – and the Turkish Cypriot north.
Turkey has stationed tens of thousands of troops in the north of the island since its 1974 invasion, which followed a coup engineered by military rulers in Athens.